This is a more precise method that is less subject to the types
of errors associated with the graphical method. It requires knowing
the x,y coordinates and heads at three locations A, B and C that
are in the x,y plane and that are not along a single line (i.e.
they form a triangle). Two additional locations D and E are then
identified along one or more of the lines forming the sides of
the triangle ABC. One of these must have the same x coordinate
as one of the three original locations and the other must have
the same y coordinate as one of the original locations. It is
then possible to determine the remaining coordinates and the heads
at these two locations using an understanding of the principles
of similar right triangles.

The example below is equivalent to that demonstrated in class,
although the figure is slightly different. Consider the case where
heads and coordinates (x,y) are known at three points A, B, and
C (shown on the figure below).

We first identify two additional points, D and E, such that
xD = xA and yE
= yB. These points
are shown on the figure below.

We can then determine the y coordinate for D using the relationship

(yD
- yC)/(yB - yC) = (xD
- xC)/(xB - xC)

Similarly, we can determine the x coordinate for E using the
relationship

(xE
- xC)/(xA - xC) = (yE
- yC)/(yA - yC)

We can then determine the heads at D and E from the relationships

(hD
- hC)/(hB - hC) = (xD
- xC)/(xB - xC)

(hE
- hC)/(hA - hC) = (yE
- yC)/(yA - yC)

The hydraulic gradient in the x direction is then computed
as

= (hB - hE)/(xB
- xE)

and that in the y direction as

= (hA - hD)/(yA
- yD)

These two hydraulic gradient vectors are then added together
to get a resultant gradient vector with magnitude

and direction

where a is an angle measured counterclockwise
from the positive x-axis (if both gradients are in the positive
x and y directions).

II. Definition of the water table

The definitions of hydraulic head, elevation head and pressure
head allow us to define the water table as the location at which
the following relations are true

a. Hydraulic head is equal to the elevation: h = z

b. The pressure is zero gage: p = po
(atmospheric pressure)

c. The pressure head is zero: p/rwg = 0

III. Darcy's K

The proportionality constant in
Darcy's Law, K, is known as the hydraulic conductivity and has
units of L/T. The hydraulic conductivity is a function of both
the properties of the porous medium that contribute to resistance
to flow (i.e. the surface area of solids per unit volume of water),
but also of the properties of the fluid itself, since internal
friction in the fluid also causes mechanical energy to be converted
to heat during flow. The properties of the porous medium alone
are quantified by the "intrinsic permeability" k. Hydraulic
conductivity is directly proportional to k. The relevant property
of the fluid is its viscosity, mw, and K is inversely proportional
to viscosity. Finally, since head is mechanical energy per unit
weight of fluid, we need to put the unit weight of the fluid,
gw = rwg,
back into K so that we actually have mechanical energy in Darcy's
Law. The result is the following definition

There are also a number of empirical
correlations that have been proposed to allow estimation of K
for granual porous media using measured values of porosity or
grain size. Some of these are listed in Table 3.5 of S&Z.
Useful parameters of a grain size distribution, as illustrated
in Figure 3.7 of S&Z, include the median diameter, d50,
the "effective diameter", d10 (10% of the grains are finer than this
diameter), and the log standard deviation of the grain size distribution.

IV. Ranges of K for geologic materials

Table 3.4 of S&Z lists ranges
of K for common geologic materials (both sediments and rocks).
Note the very large range of values (at least 12 orders of magnitude).

V. Direct measures of K via laboratory
permeameters

A. Constant Head Permeameters

Figure 3.8a in S&Z is a diagram of a constant head permeameter,
similar to those that will be used in lab. Note that the length
h indicated on this diagram actually corresponds to the change
in head (Dh) across the
column. This change in head corresponds to hin
- hout. We have assumed that there is no
loss of head through the porous plates that enclose the sediment
filled portion of the column, and no change in head in the water
that fills the reservoirs and tubes outside of the sediment filled
column. Thus, the head at the inlet (bottom) end of the column
is equal to the head at the water surface in the reservoir and
the head at the column outlet is equal to the head at the water
surface above the column. In the case of the columns we will use
in lab, the entire reservoir above the column fills with water,
so that the free surface where water and air are in contact (and
at which the pressure head p/rwg is equal to 0) occurs at the opening
of the downward pointing spout.

Results of the constant head permeameter experiment can be
analyzed directly by rearranging Darcy's law

Q = -KA(hin-hout)/(zin-zout) = +(KA)D h/L

to solve for K as

K = (Q/A)L/D h

Note that L is positive (by definition) while (zin-zout) is negative for the column set-up shown
in the text. Hence the change in sign in Darcy's Law above.

B. Falling head permeameter

A diagram of a falling head permeameter is shown in Figure
3.8b of S&Z. As in Figure 3.8a, the quantities labeled as
ho and h1should actually be shown as Dho and Dh1 since these represent changes in head
across the column at times to and t1.

The text provides an equation for the falling head permeameters.
This equation was derived by combining Darcy's law for flow through
the column with an expression for the instantaneous flow rate
out of the burette. At any instant in time, the flow rate through
the column is assumed to be steady so that

Qin = Qout (or
eqivalently DStorage = 0)

The flow rate out of the column is quantified using Darcy's
law as in the case of a constant head permeameter, but now Dh is a function of time

Qout = (KA)(Dh/L)

We can derive an equation for water flowing into the column
by recognizing that flow through the burette that serves as the
inlet reservoir can be quantified by

Qin = - a (dhin/dt)

where hin is the head at the inlet boundary
of the column (equal to the water level in the burette) and a
is the cross sectional area of the burette. Note that by this
definition, a decrease in head results in a positive Q, which
is what we want to compare to the positive (upward) Q through
the column. Also, since the head at the column outlet does not
change over time

dhin/dt = dDh/dt

The "heads" ho and h1 shown
on the figure for the falling head permeameter in the text are
actually changes in head, Dho and Dh1,
at times to and t1
respectively.

Setting the flow rate through the column equal to the flow
rate out of the burette

-a (dDh/dt)
= KA (Dh/L)

This is an ordinary differential equation that can be solved
by separation of variables